Monte Carlo Confidence Intervals for European Call Prices

Summary

The document explains how to estimate a confidence interval for a European call price using Monte Carlo simulation. The recommended procedure is to calculate the option payoff in each scenario, including zero payoffs when the underlying finishes below the strike, then compute the sample mean and standard deviation of those payoffs. The interval for the estimated payoff mean uses the standard error, and the bounds are discounted to present value.

The alternative of constructing an interval for the terminal underlying price and converting its endpoints into payoffs is rejected because the payoff function is nonlinear, so applying it to price bounds does not preserve the desired confidence interval. The response attributes this issue to Jensen’s inequality. The stated interval relies on asymptotic normality; its coverage may be poor for some simulation sizes, options, and parameter choices. The document gives a methodological answer but no empirical comparison or guidance on finite-sample corrections.

Key ideas

• Estimate the call price interval from simulated option payoffs rather than terminal asset prices.
• Include zero payoffs for scenarios in which the asset finishes below the strike.
• Use the payoff sample’s standard error to form an asymptotic confidence interval, then discount it.
• The terminal-price interval approach fails for the nonlinear payoff transformation.
• Coverage can be inaccurate for some scenario counts and option parameters.

Tags

# How to calculate confidence interval for option price?


# How to calculate confidence interval for option price?












I model option prices for European call using Monte Carlo method. What is the proper way to calculate the confidence interval?

A. -> Calculate the payoffs (there will be number of zeros as some prices go below strike) -> calculate mean and st.dev. of the payoffs -> apply the formula for the confidence interval: mean option payoff +/- z*(st.dev option payoff / sqrt(number of simulation) ) -> discount

or B. -> Calculate the mean and st.dev. of all the prices at maturity -> apply the formula for the confidence interval: mean underlying priceT +/- z*(st.dev underlying priceT / sqrt(number of simulation) ) -> calculate upper and lower band of the payoffs -> discount

where payoff is max(underlying asset priceT - option strike price,0)

## Answer by g g (score 2, accepted)

https://quant.stackexchange.com/a/22426

The right way is approach A. Approach B does not work because of the Jensen inequality. Take note that your confidence intervals are only asymptotically correct. Depending on the number of scenarios, your options and their parameters these might provide bad coverage.

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